A continuum model for alignment of self-propelled particles with anisotropy and density-dependent parameters

TL;DR

This paper extends a macroscopic model of self-propelled particles by incorporating density-dependent relaxation and anisotropic observation, demonstrating robustness of the model and analyzing conditions affecting hyperbolicity.

Contribution

It introduces modifications to the existing model to include local density effects and anisotropy, and develops a method to analyze the resulting coefficients and hyperbolicity.

Findings

Model coefficients depend on local density.

Anisotropy can lead to loss of hyperbolicity.

Asymptotic expansions reveal influence of observation angle.

Abstract

We consider the macroscopic model derived by Degond and Motsch from a time-continuous version of the Vicsek model, describing the interaction orientation in a large number of self-propelled particles. In this article, we study the influence of a slight modification at the individual level, letting the relaxation parameter depend on the local density and taking in account some anisotropy in the observation kernel (which can model an angle of vision). The main result is a certain robustness of this macroscopic limit and of the methodology used to derive it. With some adaptations to the concept of generalized collisional invariants, we are able to derive the same system of partial differential equations, the only difference being in the definition of the coefficients, which depend on the density. This new feature may lead to the loss of hyperbolicity in some regimes. We provide then a general method which enables us to get asymptotic expansions of these coefficients. These expansions shows, in some effective situations, that the system is not hyperbolic. This asymptotic study is also useful to measure the influence of the angle of vision in the final macroscopic model, when the noise is small.